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Determination of the solutions of boundary value problems for stationary stokes and Navier-Stokes equations having an unbounded Dirichlet integral

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Abstract

In unbounded domains Ω of the three-dimensional Euclidean space, having several outlets Ω i i=1,...,N, at infinity, one investigates the time-dependent solutions of the Stokes and Navier-Stokes system of equations for incompressible fluids, equal to zero at the boundary of the domain Ω and having arbitrary flows ∝ i through each of the outlets Ωi (the numbers ∝ i satisfy only the necessary condition:\(\sum\limits_{i = 1}^N {\alpha _i } = 0\)). For these solutions one establishes Phragmén-Lindelöf and Saint-Venant type theorems characterizing the growth of solutions at infinity. On their basis, one formulates well-posed boundary value problems for the above indicated systems and domain Ω and one proves their solvability for any quantities ∝ i . One investigates various properties of these solutions and one gives sufficient conditions for uniqueness theorems. In particular, when Ω is a pipe with cylindrical ends, then our solutions approach the Poiseuille flows with a given flow ∝ i , for any in the case of the Stokes system and for ∝ i smaller in absolute value than some critical value ∝ * i , in the case of the Navier-Stokes system.

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Additional information

The results of this paper have been communicated at the All-Union Conference on Nonlinear Problems of Hydrodynamics and Related Questions (Leningrad, LOMI, April 1979) and also at the XIV Symposium on Contemporary Problems and Methods of Hydrodynamics (Poland, Blazheevko, September 1979) and at the International Working Group of Nonlinear and Turbulent Processes in Physics (Kiev, September, 1979).

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 96, pp. 117–160, 1980.

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Ladyzhenskaya, O.A., Solonnikov, V.A. Determination of the solutions of boundary value problems for stationary stokes and Navier-Stokes equations having an unbounded Dirichlet integral. J Math Sci 21, 728–761 (1983). https://doi.org/10.1007/BF01094437

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  • DOI: https://doi.org/10.1007/BF01094437

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